We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Weighted Analogue of Inverse Maxwell Distribution with ApplicationsPremier Publishers
In the present study, we established a new statistical model named as weighted inverse Maxwell distribution (WIMD). Its several statistical properties including moments, moment generating function, characteristics function, order statistics, shanon entropy has been discussed. The expression for reliability, mode, harmonic mean, hazard rate function has been derived. In addition, it also contains some special cases that are well known. Moreover, the behavior of probability density function (p.d.f) has been shown through graphs by choosing different values of parameters. Finally, the performance of the proposed model is explained through two data sets. By which we conclude that the established distribution provides better fit.
Existence and Controllability Result for the Nonlinear First Order Fuzzy Neut...ijfls
In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are obtained by using the contraction principle theorem. An example to illustrate the theory.
https://arxiv.org/abs/1905.01927
A quantum mechanical model that realizes the Z2xZ2-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.
We first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Weighted Analogue of Inverse Maxwell Distribution with ApplicationsPremier Publishers
In the present study, we established a new statistical model named as weighted inverse Maxwell distribution (WIMD). Its several statistical properties including moments, moment generating function, characteristics function, order statistics, shanon entropy has been discussed. The expression for reliability, mode, harmonic mean, hazard rate function has been derived. In addition, it also contains some special cases that are well known. Moreover, the behavior of probability density function (p.d.f) has been shown through graphs by choosing different values of parameters. Finally, the performance of the proposed model is explained through two data sets. By which we conclude that the established distribution provides better fit.
Existence and Controllability Result for the Nonlinear First Order Fuzzy Neut...ijfls
In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are obtained by using the contraction principle theorem. An example to illustrate the theory.
https://arxiv.org/abs/1905.01927
A quantum mechanical model that realizes the Z2xZ2-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.
We first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
On Continuous Approximate Solution of Ordinary Differential EquationsWaqas Tariq
In this work the problem of continuous approximate solution of the ordinary differential equations will be investigated. An approach to construct the continuous approximate solution, which is based on the discrete approximate solution and the spline interpolation, will be provided. The existence and uniqueness of such continuous approximate solution will be pointed out. Its error will be estimated and its convergence will be considered. Finally, with the aid of modern PC and nathematical software three practical computer approaches to perform above construction will be offered.
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
Fixed Point Theorems for Weak K-Quasi Contractions on a Generalized Metric Sp...IJERA Editor
In this paper we obtain conditions for a k- quasi contraction on a generalized metric space with a partial order to have a fixed point. Using this, we derive certain known results as corollaries.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of 2<=m<n and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m we obtain a tower of higher order (as differential operators) even Hamiltonians, while for m odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.
https://arxiv.org/abs/2406.02188
We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU(2) using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary Σ-matrices introduced here play a role similar to ordinary matrix units, and their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The presentation of n-ary SU(2) in terms of full Σ-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q>4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4q(n-1)+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σ^het-matrices of order (4q(n-1))^4. Some examples of the lowest arities are presented.
https://arxiv.org/abs/2403.19361. *) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures, https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the unitless ternary division algebra of imaginary "half-octonions" we have introduced is ternary alternative.
https://arxiv.org/abs/2312.01366
https://www.amazon.com/s?k=duplij
https://arxiv.org/abs/2312.01366.
We introduce a new class of division algebras, hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the proposed earlier matrix polyadization procedure which increases the algebra dimension. The obtained algebras obey the binary addition and nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative and the corresponding map is n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. Then we obtain another series of n-ary algebras corresponding to the binary division algebras which have more dimension, that is proportional to intermediate arities, and they are not isomorphic to those obtained by the previous constructions. Second, we propose a new iterative process (we call it "imaginary tower"), which leads to nonunital nonderived ternary division algebras of half dimension, we call them "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the introduced unitless ternary division algebra of imaginary "half-octonions" is ternary alternative.
178 pages, 6 Chapters. DOI: 10.1088/978-0-7503-5281-9. This book presents new and prospective approaches to quantum computing. It introduces the many possibilities to further develop the mathematical methods of quantum computation and its applications to future functioning and operational quantum computers. In this book, various extensions of the qubit concept, starting from obscure qubits, superqubits and other fundamental generalizations, are considered. New gates, known as higher braiding gates, are introduced. These new gates are implemented as an additional stage of computation for topological quantum computations and unconventional computing when computational complexity is affected by its environment. Other generalizations are considered and explained in a widely accessible and easy-to-understand way. Presented in a book for the first time, these new mathematical methods will increase the efficiency and speed of quantum computing.Part of IOP Series in Coherent Sources, Quantum Fundamentals, and Applications. Key features • Provides new mathematical methods for quantum computing. • Presents material in a widely accessible way. • Contains methods for unconventional computing where there is computational complexity. • Provides methods to increase speed and efficiency. For the light paperback version use MyPrint service here: https://iopscience.iop.org/book/mono/978-0-7503-5281-9, also PDF, ePub and Kindle. For the libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Innovative-Quantum-Computing/?K=9780750352796. Amazon ordering: https://www.amazon.de/gp/product/0750352795
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories.
https://www.mdpi.com/books/book/6455
We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.
https://arxiv.org/abs/2212.08606
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Книга «Поэфизика души» представляет собой полное, на момент издания 2022 г., собрание прозаических произведений автора. Как рассказы, так и миниатюры на полстраницы, пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга поэтическими образами, воплощенными в прозе. Также включены юмористические путевые заметки о поездке в Китай.
Книга "Поэфизика души", Степан Дуплий – полное собрание прозы 2022, 230 стр. вышла в Ridero: https://ridero.ru/books/poefizika_dushi и Kindle Edition file на Амазоне: https://amazon.com/dp/B0B9Y4X4VJ . "Бумажную" книгу можно заказать на Озоне https://ozon.ru/product/poefizika-dushi-682515885/?sh=XPu-9Sb42Q и на ЛитРес: https://litres.ru/stepan-dupliy/poefizika-dushi-emocionalnaya-proza-kitayskiy-shtrih-punktir . Google books: https://books.google.com/books?id=9w2DEAAAQBAJ .
Книгу можно заказать из-за рубежа на AliExpress: https://aliexpress.com/item/1005004660613179.html .
Книга «Гравитация страсти» представляет собой полное собрание стихотворений автора на момент издания (август, 2022). Стихотворения пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга необычными поэтическими образами.
Книга "Гравитация страсти", Степан Дуплий - полное собрание стихотворений 2022, 338 стр. вышла в Ridero: https://ridero.ru/books/gravitaciya_strasti
. Книга в мягкой обложке доступна для заказа на Ozon.ru: https://ozon.ru/product/gravitatsiya-strasti-707068219/?oos_search=false&sh=XPu-9TbW9Q
, на Litres.ru: https://www.litres.ru/stepan-dupliy/gravitaciya-strasti-stihotvoreniya , за рубежом на AliExpress: https://aliexpress.com/item/1005004722134442.html , и в электронном виде Kindle file на Amazon.com: https://amazon.com/dp/B0BDFTT33W .
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in the block-diagonal matrix form (Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms in a special way to get a general shape of semisimple nonderived polyadic structures. We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.
In book: S. Duplij, "Polyadic Algebraic Structures", 2022, IOP Publishing (Bristol), Section 1.5. See https://iopscience.iop.org/book/978-0-7503-2648-3
https://arxiv.org/abs/2206.14840
The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to 'generalize' it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the 'concrete way' and provide consequent 'polyadization' of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang–Baxter equation, find its constant solutions, and introduce polyadic tensor categories.
Suitable for university students of advanced level algebra courses and mathematical physics courses.
Key features
• Provides a general, unified approach
• Widens readers perspective of the possibilities to develop standard algebraic structures
• Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
• Presents applications of 'polyadization' approach to concrete algebraic structures
https://arxiv.org/abs/2201.08479
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be "entangled" such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce two external products: 1) the iterated direct product which is componentwise, but can have arity different from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), when all multipliers are zeroless fields, which can lead to a new category of polyadic fields. Many illustrative concrete examples are presented.
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/ fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and ?-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" ?-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and ?-commutativity are studied.
We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the \textquotedblleft product\textquotedblright\ obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the \textquotedblleft Kronecker\textquotedblright\ obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.
Online first: https://www.intechopen.com/online-first/obscure-qubits-and-membership-amplitudes
Abstract: In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
https://arxiv.org/abs/2011.04370
A concept of quantum computing is proposed which naturally incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultaneously characterized by a quantum probability and a membership function. Along with the quantum amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum probability only, while the membership function can be computed through the membership amplitudes according to a chosen model. Two different versions are given here: the "product" obscure qubit in which the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations can be performed independently (i.e. quantum computation alongside truth). The measurement and entanglement of obscure qubits are briefly described.
https://arxiv.org/abs/2006.07865
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and e-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" e-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and e-commutativity are studied.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
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Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
2. Universe 2021, 7, 379 2 of 20
receive polyadic idempotents. Therefore the exchange of the commuting of idempotents
by commuting of shifts [22] is still valid.
To get polyadic idempotents (by analogy with the ordinary regularity for semigroups)
we introduce the so called sandwich regularity for polyadic semigroups (which differs
from [16]). In trying to connect the commutation of idempotents with uniqueness of inverse
elements (as in the standard regularity) in order to obtain inverse polyadic semigroups, we
observe that this is prevented by the middle elements in the sandwich polyadic regularity
conditions. Thus, further investigations are needed to develop the higher regular and
inverse polyadic semigroups.
2. Generalized n-Regular Elements in Semigroups
Here we formulate some basic notions in terms of tuples and the “quantity/number
of multiplications”, in such a way that they can be naturally generalized from the binary to
polyadic case [23].
2.1. Binary n-Regular Single Elements
Indeed, if S is a set, its `th Cartesian product S` =
`
z }| {
S × S × . . . × S consists of all
`-tuples x(`) = (x1, x2, . . . , x`), xi ∈ S (we omit powers (`) in the obvious cases x(`) ≡ x,
` ∈ N). Let S2 ≡ Sk=2 = hS | µ2, assoc2i be a binary (having the arity k = 2) semigroup
with the underlying set S, and the (binary k = 2) multiplication µ2 ≡ µk=2 : S × S → S
(sometimes we will denote µ2[x1, x2] = x1x2 ∈ SS). The (binary) associativity assoc2 :
(x1x2)x3 = x1(x2x3), ∀xi ∈ S, allows us to omit brackets, and in the language of `-
tuples x(`) the product of ` elements x1x2 . . . x` = µ
[`µ]
2
h
x(`)
i
= x(`) ∈
`
z }| {
SS . . . S, where
`µ is the “number of multiplications” (for the binary multiplication `µ = ` − 1, i.e., the
product of ` elements contains ` − 1 (binary) multiplications, whereas for polyadic (k-ary)
multiplications this is different [20]). If the number of elements is not important or evident,
we will write for the product x(`) ≡ x ∈
`
z }| {
SS . . . S, and so distinguish between the product
of ` elements x(`) ∈
`
z }| {
SS . . . S and the `-tuples x(`) ∈ S` =
`
z }| {
S × S × . . . × S. If all elements in
a `-tuple coincide, we write x` = µ
[`µ]
2
h
x(`)
i
(this form will be important in the polyadic
generalization below). An element x ∈ S satisfying x` = x (or µ
[`µ]
2
h
x(`)
i
= x) is called
(binary) `-idempotent.
An element x of the semigroup S2 is called (von Neumann) regular, if the equation
xyx = x, or (1)
µ
[`µ=2]
k=2 [x, y, x] = x, x, y ∈ S, (2)
has a solution y in S2, which need not be unique. An element y is called an inverse
element [2] (the superceded terms for y are: a reciprocal element [24], a generalized
inverse [25], a regular element conjugated to x [3]). The set of the inverse elements of x is
denoted by Vx [4].
In search of generalizing the regularity condition (1) in S2 for the fixed element x ∈ S,
we arrive at the following possibilities:
(i) Higher regularity generalization: instead of one element y ∈ S, use an n-tuple y(n) ∈ Sn
and the corresponding product y(n) ∈
n
z }| {
SS . . . S.
(ii) Higher arity generalization: instead of the binary product µk=2, consider the polyadic
(or k-ary semigroup) product with the multiplication µk :
k
z }| {
S × S × . . . × S → S.
3. Universe 2021, 7, 379 3 of 20
It follows from the higher regularity generalization (i) applied to for (1)–(2).
Definition 1. An element x of the (binary) semigroup S2 is called higher n-regular, if there exists
(at least one, not necessarily unique) (n − 1)-element solution as (n − 1)-tuple y(n−1) of
xy(n−1)
x = x, or (3)
µ
[n]
k=2
h
x, y(n−1)
, x
i
= x, x, y1, . . . , yn−1 ∈ S. (4)
Definition 2. The (n − 1)-tuple y(n−1) from (4) is called a higher n-inverse to x, or y(n−1) is an
n-inverse (tuple) for x.
Without any additional conditions, for instance splitting the (n − 1)-tuple y(n−1) for
some reason (as below), the definition (3) reduces to the ordinary regularity (1), because the
binary semigroup is closed with respect to the multiplication of any number of elements,
and so there exists z ∈ S, such that y(n−1) = z for any y1, . . . , yn−1 ∈ S. Therefore, we have
the following reduction
Assertion 1. Any higher n-regular element in a binary semigroup is 2-regular (or regular in the
ordinary sense (1)).
Indeed, if we consider the higher 3-regularity condition x1x2x3x1 = x1 for a single
element x1, and x2, x3 ∈ S, it obviously coincides with the ordinary regularity condition
for the single element x (1). However, if we cycle both the regularity conditions, they will
not necessarily coincide, and new structures can appear. Therefore, using several different
mutually consistent n-regularity conditions (3) we will be able to construct a corresponding
(binary) semigroup which will not be reduced to an ordinary regular semigroup (see
below).
2.2. Polyadic n-Regular Single Elements
Following the higher arity generalization (ii), we introduce higher n-regular single
elements in polyadic (k-ary) semigroups and show that the definition of regularity for
higher arities cannot be done similarly to ordinary regularity, but requires a polyadic
analog of n-regularity for a single element of a polyadic semigroup. Before proceeding we
introduce.
Definition 3 (Arity invariance principle). In an algebraic universal system with operations of
different arities the general form of expressions should not depend on numerical arity values.
For instance, take a binary product of 3 elements (xy)z ≡ µ2[µ2[x, y], z], its ternary ana-
log will not be µ3[x, y, z], but the expression of the same operation structure µ3[µ3[x, y, t], z, u]
which contains 5 elements.
This gives the following prescription for how to generalize expressions from the binary
shape to a polyadic shape:
(i) Write down an expression using the operations manifestly.
(ii) Change arities from the binary values to the needed higher values.
(iii) Take into account the corresponding changes of tuple lengths according to the concrete
argument numbers of operations.
Example 1. If on a set S one defines a binary operation (multiplication) µ2 : S × S → S, then one
(left) neutral element e ∈ S for x ∈ S is defined by µ2[e, x] = x (usually ex = x). However, for
4-ary operation µ4 : S4 → S it becomes µ4[e1, e2, e3, x] = x, where e1, e2, e3 ∈ S, so we can have a
neutral sequence e = e(3) = (e1, e2, e3) (a tuple e of the length 3, triple), and only in the particular
case e1 = e2 = e3 = e, do we obtain one neutral element e. Obviously, the composition of two
4. Universe 2021, 7, 379 4 of 20
(`µ = 2) 4-ary (k = 4) operations, denoted by µ
[2]
4 ≡ µ
[`µ=2]
k=4 (as in (2)) should act on a word of a
length ` = 7, since µ4[µ4[x1, x2, x3, x4], x5, x6, x7], xi ∈ S.
In general, if a polyadic system hS | µki has only one operation µk (k-ary multiplica-
tion), the length of words is “quantized”
L
(`µ)
k = `µ(k − 1) + 1, (5)
where `µ is a “number of multiplications”.
Let Sk = hS | µk, assocki be a polyadic (k-ary) semigroup with the underlying set S, and
the k-ary multiplication µk : Sk → S [20]. The k-ary (total) associativity assock can be treated
as invariance of two k-ary products composition µ
[2]
k with respect to the brackets [23]
assock : µ
[2]
k [x, y, z] = µk[x, µk[y], z] = invariant, ∀xi, yi, zi ∈ S, (6)
for all placements of the inner multiplication µk in the r.h.s. of (6) with the fixed ordering
of (2k − 1) elements (following from (5)), and x, y, z are tuples of the allowed by (5) lengths,
such that the sum of their lengths is (2k − 1). The total associativity (6) allows us to
write a product containing `µ multiplications using external brackets only µ
[`µ]
k [x] (the long
product [26]), where x is a tuple of length `µ(k − 1) + 1, because of (5). If all elements of the
`-tuple x = x(`) are the same we write it as x(`). A polyadic power (reflecting the “number of
multiplications”, but not of the number of elements, as in the binary case) becomes
xh`µi = µ
[`µ]
k
h
x`µ(k−1)+1
i
= µ
[`µ]
k
h
x(`)
i
, (7)
because of (5). So the ordinary (binary) power p (as a number of elements in a product xp)
is p = `µ + 1.
Definition 4. An element x from a polyadic semigroup Sk is called generalized polyadic
`µ
-
idempotent, if
xh`µi = x, (8)
and it is polyadic idempotent, if [20]
xh1i
≡ xh`µ=1i = µk
h
x(k)
i
= x. (9)
In the binary case, (9) and (8) correspond to the ordinary idempotent x2 = x and to
the generalized p-idempotent xp = x, where p = `µ + 1.
Recall also that a left polyadic identity e ∈ S is defined by
µk
h
e(k−1)
, x
i
= x, ∀x ∈ S, (10)
and e is a polyadic identity, if x can be on any place. If e = e(x) depends on x, then we call it
a local polyadic identity (see [27] for the binary case), and if e depends on several elements of
S, we call it a generalized local polyadic identity. It follows from (10) that all polyadic identities
(or neutral elements) are polyadic idempotents (9).
An element x of a k-ary semigroup Sk is called regular [22], if the equation (cf. (2))
µ
[2]
k
h
x, y(2k−3)
, x
i
= x, x ∈ S, (11)
5. Universe 2021, 7, 379 5 of 20
has 2k − 3 solutions y1, . . . , y2k−3 ∈ S, which need not be unique (since L
(2)
k − 2 = 2k − 3,
see (5)). If we have only the relation (11), then polyadic associativity allows us to apply one
(internal) polyadic multiplication and remove k − 1 elements to obtain [22]
µk
h
x, z(k−2)
, x
i
= x, x ∈ S, (12)
and we call (12) the reduced polyadic regularity of a single element, while (11) will be called
the full regularity of a single element. Thus, for a single element x ∈ S, by analogy with
Assertion 1, we have
Assertion 2. In a polyadic semigroup Sk, if any single element x ∈ S satisfies the full regularity
(11), it also satisfies the reduced regularity (12).
From (12) it follows:
Corollary 1. Reduced regularity exists, if and only if the arity of multiplication is strongly more
than binary, i.e., k ≥ 3.
Example 2. In the minimal ternary case k = 3 we have the regularity µ
[2]
3 [x1, x2, x3, x4, x1] =
x1, xi ∈ S, and putting z = µ3[x2, x3, x4] gives the reduced regularity
µ3[x1, z, x1] = x1. (13)
However, these regularities will give different cycles: one is of length 5, and the other is of
length 3.
We now construct a polyadic analog of the higher n-regularity condition (4) for a
single element.
Definition 5. An element x of a polyadic semigroup Sk is called full higher n-regular, if there
exists (at least one, not necessarily unique) (n(k − 1) − 1)-element solution (tuple) y(n(k−1)−1) of
µ
[n]
k
h
x, y(n(k−1)−1)
, x
i
= x, x, y1, . . . , yn(k−1)−1 ∈ S, n ≥ 2. (14)
Definition 6. The tuple y(n(k−1)−1) from (14) is called higher n-regular k-ary inverse to x, or
y(n(k−1)−1) is n-inverse k-ary tuple for x of the length (n(k − 1) − 1).
Example 3. In the full higher 3-regular ternary semigroup S3 = hS | µ3i we have for each element
x ∈ S the condition
µ
[2]
3 [x, y1, y2, y3, y4, y5, x] = x, (15)
and so each element x ∈ S has a 3-inverse ternary 5-tuple y(5) = (y1, y2, y3, y4, y5), or the 5-tuple
y(5) is the higher 3-regular ternary inverse of x.
Comparing (14) with the full regularity condition (11) and its reduction (12), we
observe that the full n-regularity condition can be reduced several times to get different
versions of reduced regularity for a single element.
Definition 7. An element x of a polyadic semigroup Sk is called m-reduced higher n-regular, if there
exists a (not necessarily unique) ((n − m)(k − 1) − 1)-element solution (tuple) y((n−m)(k−1)−1) of
µ
[n]
k
h
x, y((n−m)(k−1)−1)
, x
i
= x, x, y1, . . . , y(n−m)(k−1)−1 ∈ S, 1 ≤ m ≤ n − 1, n ≥ 2. (16)
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Example 4. In the ternary case k = 3, we have (for a single element x1 ∈ S) the higher 3-regularity
condition
µ
[3]
3 [x1, x2, x3, x4, x5, x6, x1] = x1, xi ∈ S. (17)
We can reduce this relation twice, for instance, as follows: put t = µ3[x2, x3, x4] to get
µ
[2]
3 [x1, t, x5, x6, x1] = x1, (18)
and then z = µ3[t, x5, x6] and obtain
µ3[x1, z, x1] = x1, t, z ∈ S, (19)
which coincides with (13). Again we observe that (17)–(19) give different cycles of the length 7, 5
and 3, respectively.
3. Higher n-Inverse Semigroups
Here we introduce and study the higher n-regular and n-inverse (binary) semigroups.
We recall the standard definitions to establish notations (see, e.g., [2,3,5]).
There are two different definitions of a regular semigroup:
(i) One-relation definition. (20)
(ii) Multi-relation definition . (21)
According to the one-relation definition (i) (20): A semigroup S ≡ S2 = hS | µ2, assoc2i
is called regular if each element x ∈ S has its inverse element y ∈ S defined by xyx = x (1),
and y need not be unique [28].
Elements x1, x2 of S are called inverse to each other [2] (or regular conjugated [3],
mutually regular), if
x1x2x1 = x1, x2x1x2 = x2, (22)
µ
[2]
2 [x1, x2, x1] = x1, µ
[2]
2 [x2, x1, x2] = x2, x1, x2 ∈ S. (23)
According to the multi-relation definition (ii) [29] (21): A semigroup is called regular,
if each element x1 has its inverse x2 (22), and x2 need not be unique.
For a binary semigroup S2 and ordinary 2-regularity these definitions are equiva-
lent [2,24]. Indeed, if x1 has some inverse element y ∈ S, and x1yx1 = x1, then choosing
x2 = yx1y, we immediately obtain (22).
Definition 8. A 2-tuple X(2) = (x1, x2) ∈ S × S is called 2-regular sequence of inverses, if it
satisfies (22).
Thus we can define regular semigroups in terms of regular sequences of inverses.
Definition 9. A binary semigroup is called a regular (or 2-regular) semigroup, if each element
belongs to a (not necessary unique) 2-regular sequence of inverses X(2) ∈ S × S.
3.1. Higher n-Regular Semigroups
Following [16], in a regular semigroup S2, for each x1 ∈ S there exists x2 ∈ S such
that x1x2x1 = x1, and for x2 ∈ S there exists y ∈ S satisfying and x2yx2 = x2. Denoting
x3 = yx2, then x2 = x2(yx2) = x2x3, and we observe that x1x2x3x1 = x1, i.e., in a regular
semigroup, from 2-regularity there follows 3-regularity of S2 in the one-relation definition
(i) (cf. for a single element Assertion 1). By analogy, in a regular semigroup, 2-regularity
of an element implies its n-regularity in the one-relation definition (i) (20). Indeed, in this
point there is a difference between the definitions (i) (20) and (ii) (21).
Let us introduce higher analogs of the regular sequences (22) (see Definition 8).
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Definition 10. A n-tuple X(n) = (x1, . . . , xn) ∈ Sn is called an (ordered) n-regular sequence of
inverses, if it satisfies the n relations
x1x2x3 . . . xnx1 = x1, (24)
x2x3 . . . xnx1x2 = x2, (25)
.
.
.
xnx1x2 . . . xn−1xn = xn, (26)
or in the manifest form of the binary multiplication µ2 (which is needed for the higher arity
generalization)
µ
[n]
2 [x1, x2, x3, . . . , xn, x1] = x1, (27)
µ
[n]
2 [x2, x3, . . . , xn, x1, x2] = x2, (28)
.
.
.
µ
[n]
2 [xn, x1, x2, . . . , xn−1, xn] = xn. (29)
Denote for each element xi ∈ S, i = 1, . . . , n, its n-inverse (n − 1)-tuple (see Definition 2)
by xi ∈ S(n−1), where
x1 = (x2, x3, . . . , xn−1, xn), (30)
x2 = (x3, x4 . . . , xn−1, xn, x1), (31)
.
.
.
xn = (x1, x2, . . . , xn−2, xn−1). (32)
Then the definition of an n-regular sequence of inverses (24)–(29) will take the concise
form (cf. (1) and (3))
xixixi = xi, (33)
µ
[n]
2 [xi, xi, xi] = xi, xi ∈ S, i = 1, . . . , n, (34)
where the products xi ∈
n−1
z }| {
SS . . . S and the (n − 1)-tuples xi ∈
n−1
z }| {
S × S × . . . × S (30)–(32).
Now by analogy with Definition 9 we have the following (multi-relation):
Definition 11. A binary semigroup S2 is called a (cyclic) n-regular semigroup S
n-reg
2 , if each of its
elements belongs to a (not necessary unique) n-regular sequence of inverses X(n) ∈ Sn satisfying
(24)–(26).
This definition incorporates the ordinary regular semigroups (22) by putting n = 2.
Remark 1. The proposed concept of n-regularity is strongly multi-relational, in that all the relations
in (24)–(26) or (33) should hold, which leads us to consider all the elements in the sequence of
inverses on a par and cyclic, analogously to the ordinary case (22). This can be also expressed more
traditionally as “one element of a n-regular semigroup has n − 1 inverses”.
Example 5. In the 3-regularity case we have three relations defining the regularity
x1x2x3x1 = x1, (35)
x2x3x1x2 = x2, (36)
x3x1x2x3 = x3. (37)
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Now the sequence of inverses is X(3) = (x1, x2, x3), and we can say in a symmetrical way: all
of them are “mutually inverses one to another”, if all three relations (35)–(37) hold simultaneously.
Also, using (30)–(32) we see that each element x1, x2, x3 has its 3-inverse 2-tuple (or, informally,
the pair of inverses)
x1 = (x2, x3), (38)
x2 = (x3, x1), (39)
x3 = (x1, x2). (40)
Remark 2. In what follows, we will use this 3-regularity example, when providing derivations and
proofs, for clarity and conciseness. This is worthwhile, because the general n-regularity case does
not differ from n = 3 in principle, and only forces us to consider cumbersome computations with
many indices and variables without enhancing our understanding of the structures.
Let us formulate the Thierrin theorem [24] in the n-regular setting, which connects
one-element regularity and multi-element regularity.
Lemma 1. Every n-regular element in a semigroup has its n-inverse tuple with n − 1 elements.
Proof. Let x1 be a 3-regular element x1 of a semigroup S2, then we can write
x1 = x1y2y3x1, x1, y2, y3 ∈ S. (41)
We are to find elements x2, x3 which satisfy (35)–(37). Put
x2 = y2y3x1y2, (42)
x3 = y3x1y2y3. (43)
Then, for the l.h.s. of (35)–(37) we derive
x1x2x3x1 = x1y2y3(x1y2y3x1)y2y3x1 = x1y2y3(x1y2y3x1)
= x1y2y3x1 = x1, (44)
x2x3x1x2 = y2(y3x1y2y3)(x1y2y3x1)y2y3(x1y2y3x1)y2
= y2x3(x1x2y3x1)y2 = y2x3x1y2 = x2, (45)
x3x1x2x3 = y3x1y2y3(x1y2y3x1)y2y3(x1y2y3x1)y2y3
= y3(x1y2y3x1)y2y3 = y3x1y2y3 = x3. (46)
Therefore, if we have any 3-regular element (41), the relations (35)–(40) hold.
We now show that in semigroups the higher n-regularity (in the multi-relation formu-
lation) is wider than for 2-regularity.
Lemma 2. If a semigroup S2 is n-regular, such that the n relations (33) are valid, then S2 is
2-regular as well.
Proof. In the case n = 3 we use (35)–(37) and denote x2x3 = y ∈ S. Then x1yx1 = x1, and
multiplying (36) and (37) by x3 from the right and x2 from the left, respectively, we obtain
(x2x3x1x2)x3 = (x2)x3 =⇒ (x2x3)x1(x2x3) = (x2x3) =⇒ yx1y = y, (47)
x2(x3x1x2x3) = x2(x3) =⇒ (x2x3)x1(x2x3) = (x2x3) =⇒ yx1y = y. (48)
This means that x1 and y are mutually 2-inverse (ordinary inverse [2]), and so S2 is
2-regular.
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Definition 12. A semigroup S2 is called a pure n-regular semigroup, if there no additional relations
beyond the n-regularity (24)–(26) imposed on n-regular sequence of inverses X(n) = (x1, . . . , xn).
Oppositely, a n-regular semigroup S2 with extra relations on X(n) is called impure, and these
relations are called impuring relations.
For instance, an ordinary regular (2-regular) semigroup defined by (22) only, is a pure
regular semigroup. If we add the commutativity condition x1x2 = x2x1, then we obtain an
impure regular semigroup, which is an inverse semigroup being actually a semilattice of
abelian groups (see, e.g., [6]).
The converse is connected with imposing extra relations in the semigroup S2.
Theorem 1. The following statements are equivalent for a pure regular (2-regular) semigroup S2:
(i) The pure 2-regular semigroup S2 is pure n-regular.
(ii) The pure 2-regular semigroup S2 is cancellative.
Proof. (i)⇒(ii) Suppose S2 is 2-regular, which means that there are tuples satisfying (22),
or in our notation here for each x1 ∈ S there exists y ∈ S such that x1yx1 = x1 and yx1y = y.
Let y be presented as a product of two arbitrary elements y = x2x3, x2, x3 ∈ S, which is
possible since the underlying set S of the semigroup is closed with respect to multiplication.
We now try to get the (35)–(37) without additional relations. After substitution y into the
2-regularity conditions, we obtain
x1yx1 = x1 =⇒ x1(x2x3)x1 = x1 =⇒ x1x2x3x1 = x1, (49)
yx1y = y =⇒ (x2x3)x1(x2x3) = (x2x3) =⇒ (x2x3x1x2)x3 = (x2)x3 (50)
=⇒ x2(x3x1x2x3) = x2(x3). (51)
We observe that the first line coincides with (35), but obtaining (36) and (37) from the
second and third lines here requires right and left cancellativity (by the elements outside
the brackets in the last equalities), respectively.
(ii)⇒(i) After applying cancellativity to (50) and (51), then equating expressions in
brackets one obtains (36) and (37), correspondingly. The first line (49) coincides with (35) in
any case.
It is known that a cancellative regular semigroup is a group [2,7].
Corollary 2. If a pure 2-regular semigroup S2 is pure n-regular, it is a group.
If we consider impure n-regular semigroups, these can be constructed from pure
2-regular semigroups without the cancellativity requirement, but with special additional
relations and extra idempotents. For instance, in case of an impure 3-regular semigroup
we have
Proposition 1. If a semigroup is pure 2-regular we can construct an impure 3-regular semigroup
with three impuring relations.
Proof. Let S2 be a pure 2-regular semigroup (in multi-relation definition (22)), such that
x1x3x1 = x1, x3x1x3 = x3, x1, x3 ∈ S, (52)
have at least one solution. Introduce an additional element x2 = x3x1 ∈ S, which is
idempotent (from (52)) . It is seen that the triple X(3) = {x1, x2, x3} is a 3-inverse sequence,
because (35)–(37) take place, indeed,
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x1 = x1x3x1 = x1(x3x1)x3x1 = x1x2x3x1, (53)
x2 = x3x1 = (x3x1)x3x1(x3x1) = x2x3x1x2, (54)
x3 = x3x1x3 = x3x1(x3x1)x3 = x3x1x2x3. (55)
The impuring relations
x1x2 = x1, x2x3 = x3, x3x1 = x2 (56)
follow from (52) and the definition of x2.
In the same way one can construct recursively an impure (n + 1)-regular semigroup
from a pure n-regular semigroup. Taking into account Lemma 2, we arrive at
Corollary 3. Impure n-regular semigroups are equivalent to pure 2-regular semigroups.
3.2. Idempotents and Higher n-Inverse Semigroups
Lemma 3. In an n-regular semigroup there are at least n (binary higher) idempotents (no summa-
tion)
ei = xixi, i = 1, . . . , n. (57)
Proof. Multiply (33) by xi from the right to get xixixixi = xixi.
In the case n = 3 the (higher) idempotents (57) become
e1 = x1x2x3, e2 = x2x3x1, e3 = x3x1x2, (58)
having the following “chain” commutation relations with elements
e1x1 = x1e2 = x1, (59)
e2x2 = x2e3 = x2, (60)
e3x3 = x3e1 = x3. (61)
Corollary 4. Each higher idempotent ei is a left unit (neutral element) for xi and a right unit
(neutral element) for xi−1.
Lemma 4. If in an n-regular semigroup (higher) idempotents commute, then each element has a
unique n-regular sequence of inverses.
Proof. Suppose in a 3-regular semigroup defined by (35)–(37) the element x1 has two pairs
of inverses (30) x1 = (x2, x3) and x0
1 = (x0
2, x3). Then, in addition to (35)–(37) and the triple
of idempotents (58) we have
x1x0
2x3x1 = x1, (62)
x0
2x3x1x0
2 = x0
2, (63)
x3x1x0
2x3 = x3. (64)
and
e0
1 = x1x0
2x3, e0
2 = x0
2x3x1, e0
3 = x3x1x0
2, (65)
and also the idempotents (58) and (65) commute. We derive
x2 = x2x3x1x2 = x2x3 x1x0
2x3x1
x1x2 = x2x3 x1x0
2x3 x1x0
2x3x1
x2
= x2x3x1x0
2 x3x1x0
2
(x3x1x2) = x2x3x1x0
2(x3x1x2) x3x1x0
2
(66)
= (x2x3x1) x0
2x3x1
x2x3x1x0
2 = x0
2x3x1
(x2x3x1)x2x3x1x0
2
= x0
2x3(x1x2x3x1)x2x3x1x0
2 = x0
2x3(x1x2x3x1)x0
2 = x0
2x3x1x0
2 = x0
2.
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A similar derivation can be done for x00
1 = (x2, x00
3 ) and other indices.
Definition 13. A higher n-regular semigroup is called a higher n-inverse semigroup, if each
element xi ∈ S has its unique n-inverse (n − 1)-tuple xi ∈ S(n−1) (30)–(32).
Thus, as in the regular (multi-relation 2-regular) semigroups [30–32], we arrive at a
higher regular generalization of the characterization of the inverse semigroups by
Theorem 2. A higher n-regular semigroup is an inverse semigroup, if their idempotents commute.
It is known that an inverse (2-inverse) semigroup is a group, if it contains exactly
one idempotent, and so groups are inverse semigroups (see, e.g., [2,6]). Nevertheless, for
n-inverse semigroups this is not true. For instance, if the elements x, 1 are in a group, then
3-regularity conditions (35)–(37) are satisfied by x1 = 1, x2 = x, x3 = x−1 for all x, and
therefore 2-inverses are not unique for groups. Thus, we have
Assertion 3. The n-inverse semigroups do not contain groups.
Further properties of n-inverse semigroups will be studied elsewhere using the semi-
group theory methods (see, e.g., [2,6,7,33]).
4. Higher n-Inverse Polyadic Semigroups
We now introduce a polyadic version of the higher n-regular and n-inverse semigroups
defined in the previous section. For this we will use the relations with the manifest
form of the binary multiplication to follow (ii)–(iii) of the arity invariance principle (see
Definition 3).
4.1. Higher n-Regular Polyadic Semigroups
First, define the polyadic version of the higher n-regular sequences (27)–(29) in the
framework of the arity invariance principle by substituting µ
[n]
2 7→ µ
[n]
k and then changing
sizes of tuples, and generalizing to higher n (11).
Definition 14. In a polyadic (k-ary) semigroup Sk = hS | µki (see Section 2.2) a n(k − 1)-tuple
X(n(k−1)) =
x1, . . . , xn(k−1)
∈ Sn(k−1) is called a polyadic n-regular sequence of inverses, if it
satisfies the n(k − 1) relations
µ
[n]
k
h
x1, x2, x3, . . . , xn(k−1), x1
i
= x1, (67)
µ
[n]
k
h
x2, x3, . . . , xn(k−1), x1, x2
i
= x2, (68)
.
.
.
µ
[n]
k
h
xn(k−1), x1, x2, . . . , xn(k−1)−1, xn(k−1)
i
= xn(k−1). (69)
Definition 15. A polyadic semigroup Sk is called an n-regular semigroup, if each element belongs
to a (not necessary unique) n-regular sequence of inverses X(n(k−1)) ∈ Sn(k−1).
In a polyadic n-regular semigroup, each element, instead of having one inverse
element will have a tuple, and we now, instead of (1) and (2), have
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Definition 16. In a polyadic (k-ary) n-regular semigroup each element xi ∈ S, where i =
1, . . . , n(k − 1), has a polyadic n-inverse, as the (n(k − 1) − 1)-tuple x
[k]
i ∈ S(n(k−1)−1) (cf. the
binary case Definition 2 and (30)–(32))
x
[k]
1 =
x2, x3, . . . , xn(k−1)−1, xn(k−1)
, (70)
x
[k]
2 =
x3, x4 . . . , xn(k−1)−1, xn(k−1), x1
, (71)
.
.
.
x
[k]
n(k−1)
=
x1, x2, . . . , xn(k−1)−2, xn(k−1)−1
. (72)
In this notation the polyadic (k-ary) n-regular sequence of inverses (67)–(69) can be
written in the customary concise form (see (34) for a binary semigroup)
µ
[n]
k
h
xi, x
[k]
i , xi
i
= xi, xi ∈ S, i = 1, . . . , n(k − 1). (73)
Note that at first sight one could consider here also the procedure of reducing the
number of multiplications from any number to one [22], as in the single element regularity
case (see Assertion 2 and Examples 2 and 4). However, if we consider the whole polyadic
n-regular sequence of inverses (67)–(69) consisting of n(k − 1) relations (67)–(69), it will be
not possible in general to reduce arity in all of them simultaneously.
Example 6. If we consider the ordinary regularity (2-regularity) for a ternary semigroup S3, we
obtain the ternary regular sequence of n(k − 1) = 4 elements and four regularity relations
µ
[2]
3 [x1, x2, x3, x4, x1] = x1, (74)
µ
[2]
3 [x2, x3, x4, x1, x2] = x2, (75)
µ
[2]
3 [x3, x4, x1, x2, x3] = x3, (76)
µ
[2]
3 [x4, x1, x2, x3, x4] = x4, xi ∈ S. (77)
Elements x1, x2, x3, x4 have these ternary inverses (cf. the binary 3-regularity (38)–(40))
x
[3]
1 = (x2, x3, x4), (78)
x
[3]
2 = (x3, x4, x1), (79)
x
[3]
3 = (x4, x1, x2), (80)
x
[3]
4 = (x1, x2, x3). (81)
Remark 3. This definition of regularity in ternary semigroups is in full agreement with the arity
invariance principle (Definition 3): it has (the minimum) two (k-ary) multiplications as in the
binary case (2). Although we can reduce the number of multiplications to one, as in (13) for a single
relation, e.g., by the substitution z = µ3[x2, x3, x4], this cannot be done in all the cyclic relations
(74)–(77): the third relations (76) cannot be presented in terms of z as well as the right hand sides.
Remark 4. If we take the definition of regularity for a single relation, reduce it to one (k-ary)
multiplication and then “artificially” cycle it (see [22] and references citing it), we find a conflict with
the arity invariance principle. Indeed, in the ternary case we obtain µ3[x, y, x] = x, µ3[y, x, y] =
y. Despite the similarity to the standard binary regularity (22) and (23), it contains only one
multiplication, and therefore it could be better treated as a symmetry property of the ternary product
µ3, rather than as a relation between variables, e.g., as a regularity which needs at least two products.
Also, the length of the sequence is two, as in the binary case, but it should be n(k − 1) = 4, as in
the ternary 2-regular sequence (74)–(77) according to the arity invariance principle.
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Example 7. For a 3-regular ternary semigroup S3 = hS | µ3i with n = 3 and k = 3 we have a
3-regular sequence of 6 (mutual) inverses X(6) = (x1, x2, x3, x4, x5, x6) ∈ S6, which satisfy the
following six 3-regularity conditions
µ
[3]
3 [x1, x2, x3, x4, x5, x6, x1] = x1, (82)
µ
[3]
3 [x2, x3, x4, x5, x6, x1, x2] = x2, (83)
µ
[3]
3 [x3, x4, x5, x6, x1, x2, x3] = x3, (84)
µ
[3]
3 [x4, x5, x6, x1, x2, x3, x4] = x4, (85)
µ
[3]
3 [x5, x6, x1, x2, x3, x4, x5] = x5, (86)
µ
[3]
3 [x6, x1, x2, x3, x4, x5, x6] = x6, xi ∈ S. (87)
This is the first nontrivial case in both arity k 6= 2 and regularity n 6= 2 (see Remark 2).
Remark 5. In this example we can reduce the number of multiplications, as in Example 4, to
one single relation, but not to all of them. For instance, we can put in the first relations (82)
t = µ3[x2, x3, x4] and further z = µ3[t, x5, x6], but, for instance, the third relation (84) cannot be
presented in terms of t, z, as in the first one in (82)–(87), because of the splitting of variables in t, z,
as well as in the right hand sides.
The polyadic analog of the Thierrin theorem [24] in n-regular setting is given by:
Lemma 5. Every n-regular element in a polyadic (k-ary) semigroup has a polyadic n-inverse tuple
with (n(k − 1) − 1) elements.
Its proof literally repeats that of Lemma 1, but with different lengths of sequences.
The same is true for the Lemma 2 and Theorem 1 by exchanging S2 → Sk.
Definition 17. A higher n-regular polyadic semigroup Sk is called a higher n-inverse polyadic
semigroup, if each element xi ∈ S has a unique n-inverse (n(k − 1) − 1)-tuple xi ∈ S(n(k−1)−1)
(70)–(72).
In searching for polyadic idempotents we observe that the binary regularity (34) and
polyadic regularity (73) differ considerably in lengths of tuples. In the binary case any
length is allowed, and one can define idempotents ei by (57), as a left neutral element for
xi, such that (no summation) eixi = xi , xi ∈ S2 (see (59)–(61)). At first sight, for polyadic
n-regularity (73), we could proceed in a similar way. However we have
Proposition 2. In the n-regular polyadic (k-ary) semigroup the length of the tuple (xi, xi) is
allowed to give an idempotent, only if k = 2, i.e., the semigroup is binary.
Proof. The allowed length of the tuple (xi, xi) (to give one element, an idempotent, see
(5)), where xi are in (70)–(72), is `µ(k − 1) + 1 to apply `µ multiplications. While the tuple
(xi, xi, xi) in (73) has the given length n(k − 1) + 1, and we obtain the equation for ` as
`µ(k − 1) + 1 = n(k − 1) =⇒ `µ = n −
1
k − 1
(88)
The equation (88) has only one solution over N namely `µ = n − 1, iff k = 2.
This means that to go beyond the binary semigroups k 2 and have idempotents, one
needs to introduce a different regularity condition to (11) and (73), which we will do below.
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4.2. Sandwich Polyadic n-Regularity
Here we go in the opposite direction to that above: we will define the idempotents
and then construct the needed regularity conditions using them, which in the limit k = 2
and n = 2 will give the ordinary binary regularity (22) and (23).
Let us formulate the binary higher n-regularity (34) in terms of the local polyadic
identities (10). We write the idempotents (57) in the form
ei = µ2[xi, xi], (89)
where xi are n-inverses being (n − 1)-tuples (30)–(32).
In terms of the idempotents ei (89) the higher n-regularity conditions (34) become
µ2[ei, xi] = xi. (90)
Assertion 4. The binary n-regularity conditions coincide with the definition of the local left
identities.
Proof. Compare (90) and the definition (10) with k = 2 (see [27]).
Thus, the main idea is to generalize (using the arity invariance principle) to the
polyadic case the binary n-regularity in the form (90) and also idempotents (89), but not
(33) and (34), as it was done in (73).
By analogy with (89) let us introduce
e0
i = µ
[n−1]
k
xi,
n−1
z }| {
x0
i, . . . , x0
i
, xi ∈ S, i = 1, . . . , k, (91)
where x0
i is the polyadic (k − 1)-tuple for xi (which differs from (70)–(72))
x0
1 = (x2, x3, . . . , xk), (92)
x0
2 = (x3, x4, . . . , xk, x1), (93)
.
.
.
x0
k = (x1, x2, . . . , xk−2, xk−1). (94)
Definition 18. Sandwich polyadic n-regularity conditions are defined as
µk
k−1
z }| {
e0
i, . . . , e0
i, xi
= xi, i = 1, . . . , k. (95)
Observe that the conditions (95) coincide with the definition of the generalized local
left polyadic identities (10) (see Assertion 4), and therefore we can define the sandwich
polyadic n-regularity (95) in an alternative (multi-relation (21)) way
Definition 19. A polyadic semigroup Sk is sandwich n-regular, if there exists k-tuple X(k) =
(x1, x2, x3, . . . , xk) in which each element xi has its left generalized local polyadic identity e0
i of the
form (91).
It follows from (95), that e0
i are k polyadic idempotents (9)
e0
i
h1i
= µk
k
z }| {
e0
i, . . . , e0
i
= e0
i, i = 1, . . . , k. (96)
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In the binary limit k = 2 and n = 2 (91)–(95) give the ordinary regularity (22) in the
multi-relation definition (21).
By analogy with Definition 15 we have
Definition 20. A k-ary semigroup Sk is called a sandwich polyadic n-regular semigroup Ssandw
k ,
if each element has a (not necessary unique) k-tuple of inverses satisfying (91)–(95).
Now instead of (70)–(72) we have for the inverses
Definition 21. Each element of a sandwich polyadic (k-ary) n-regular semigroup xi ∈ S, i =
1, . . . , k − 1 has its (sandwich) polyadic k-inverse, as the (k − 1)-tuple x
0[k]
i ∈ S(k−1) (see (92)–(94)).
Example 8. In the ternary case k = 3 and n = 2 we have the following 3 ternary idempotents
e0
1 = µ3[x1, x2, x3], (97)
e0
2 = µ3[x2, x3, x1], (98)
e0
3 = µ3[x3, x1, x2], xi ∈ S, (99)
such that
µ3
e0
i, e0
i, e0
i
= e0
i, i = 1, 2, 3. (100)
The sandwich ternary regularity conditions become (cf. (74)–(77))
µ
[3]
3 [x1, x2, x3, x1, x2, x3, x1] = x1, (101)
µ
[3]
3 [x2, x3, x1, x2, x3, x1, x2] = x2, (102)
µ
[3]
3 [x3, x1, x2, x3, x1, x2, x3] = x3. (103)
Each of the elements x1, x2, x3 ∈ S has a 2-tuple of the ternary inverses (cf. the inverses for
the binary 3-regularity (38)–(40) and the ternary regularity (78)–(81)) as
x0
1 = (x2, x3), (104)
x0
2 = (x3, x1), (105)
x0
3 = (x1, x2). (106)
A ternary semigroup in which the sandwich ternary regularity (101)–(103) has a solution
(starting from any of element xi) is a sandwich regular ternary semigroup Ssandw
3 .
It can be seen from (101)–(103) why we call such regularity “sandwich”: each xi
appears in the l.h.s. not only two times, on the first and the last places, as in the previous
definitions, but also in the middle, which gives the possibility for us to define idempo-
tents. In general, the ith condition of sandwich regularity (95) will contain k − 2 middle
elements xi.
Remark 6. In (101) we can also reduce the number of multiplications to one (see [22]), but only in
a single relation (see Example 4 and Remark 5). However, in this case the whole system (101)–(103)
will loose its self-consistency, since we are using the multi-relation definition of the sandwich regular
semigroup (see (22) for the ordinary regularity).
We can obtain a matrix representation of a sandwich polyadic semigroup by using the
method of antitriangle supermatrices [34] for regular semigroups and bands together with
the general form of a polyadic matrix [13].
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Example 9. Consider the sandwich regular ternary semigroup Ssandw
3 from Example 8 and its
continuous 4-parameter representation in the set M
adg
4 of 4 × 4 noninvertible matrices of the special
antitriangle form (cf. [34] for the binary case)
Mε(t, u, v, w) =
0 0 0 εt
0 0 εu 1
0 εv 0 0
εw 1 0 0
∈ M
adg
4 , (107)
where t, u, v, w ∈ C and ε is a nilpotent dual number ε2 = 0 (see, e.g., [35]). The set M
adg
4 is closed
with respect to the ordinary triple matrix product only, while the multiplication of two matrices
(107) is obviously out of the set M
adg
4 . This allows us to define the ternary product µM
3 on the set
M
adg
4 as
µM
3 [Mε(t1, u1, v1, w1), Mε(t2, u2, v2, w2), Mε(t3, u3, v3, w3)]
= Mε(t1, u1, v1, w1)Mε(t2, u2, v2, w2)Mε(t3, u3, v3, w3) = Mε(t1, u3, v1, w3). (108)
Thus, M
adg
3 =
D
M
adg
4 | µM
3
E
is a ternary semigroup, because the total ternary associativity
of µM
3 (6) is governed by the associativity of the ordinary matrix multiplication in the r.h.s. of (108).
Then the matrix representation Φsandw
3 : Ssandw
3 → M
adg
3 is given by
x 7→ Mε(t, u, v, w), x ∈ Ssandw
3 , Mε(t, u, v, w) ∈ M
adg
3 , (109)
such that putting xi 7→ Mε(ti, ui, vi, wi), i = 1, 2, 3, we obtain the sandwich ternary regularity
(101)–(103). The semigroup M
adg
3 is an example of the ternary band, because, as it follows from
(107), each element of M
adg
4 is a ternary idempotent (see (9))
(Mε(t, u, v, w))3
= Mε(t, u, v, w). (110)
Therefore, in the terminology of Definition 12, M
adg
3 is an impure regular semigroup, since it
contains an additional (to regularity) condition (110) following from the concrete matrix structure
(107), while Ssandw
3 is a pure regular semigroup, if no other conditions except the regularity (101)–
(103) are imposed. For the representation Φsandw
3 (109) to be faithful, the semigroup Ssandw
3 should
be also impure regular and idempotent (as (110)), i.e., the additional impuring conditions are
µ3[x, x, x] = x, ∀x ∈ S, which means that Ssandw
3 becomes a ternary band as well.
Remark 7. The introduced sandwich regularity (95) differs from [16] considerably. For simplicity
we consider the ternary case (101)–(103) and compare it with [16] (in our notation)
µ
[4]
3 [x1, x2, x3, x4, x1, x5, x6, x7, x1] = x1. (111)
First, we note that only the one-relation definition (20) was used in [16]. By analogy with
(12), we can provide in (111) the reduction as y1 = µ3[x2, x3, x4], y2 = µ3[x5, x6, x7] to get
µ
[2]
3 [x1, y1, x1, y2, x1] = x1. (112)
We cannot connect the sandwich ternary regularity (101) and (111)–(112), because the length
of ternary words is “quantized” (see (5)), and therefore it can be only odd L
(`µ)
3 = 2`µ + 1, `µ ∈ N,
where `µ is the number of ternary multiplications. Second, for the same reason there is no natural
way to introduce idempotents, as in (97)–(99), using (111) and (112) only.
17. Universe 2021, 7, 379 17 of 20
Example 10. The non-trivial case in both higher arity k 6= 2 and higher sandwich regularity
n 6= 2 is the sandwich 3-regular 4-ary semigroup S4 in which there exist 4 elements satisfying the
higher sandwich 3-regularity relations
µ
[4]
4 [x1, x2, x3, x4, x2, x3, x4, x1, x2, x3, x4, x2, x3, x4, x1, x2, x3, x4, x2, x3, x4, x1] = x1, (113)
µ
[4]
4 [x2, x3, x4, x1, x3, x4, x1, x2, x3, x4, x1, x3, x4, x1, x2, x3, x4, x1, x3, x4, x1, x2] = x2, (114)
µ
[4]
4 [x3, x4, x1, x2, x4, x1, x2, x3, x4, x1, x2, x4, x1, x2, x3, x4, x1, x2, x4, x1, x2, x3] = x3, (115)
µ
[4]
4 [x4, x1, x2, x3, x1, x2, x3, x4, x1, x2, x3, x1, x2, x3, x4, x1, x2, x3, x1, x2, x3, x4] = x4. (116)
The 4-ary idempotents are
e0
1 = µ4[x1, x2, x3, x4, x2, x3, x4], (117)
e0
2 = µ4[x2, x3, x4, x1, x3, x4, x1],
e0
3 = µ4[x3, x4, x1, x2, x4, x1, x2], (118)
e0
4 = µ4[x4, x1, x2, x3, x1, x2, x3], (119)
and they obey the following commutation relations with elements (cf. (59)–(61))
µ4
e0
1, e0
1, e0
1, x1
= µ4
x1, e0
2, e0
2, e0
2
= x1, (120)
µ4
e0
2, e0
2, e0
2, x2
= µ4
x2, e0
3, e0
3, e0
3
= x2, (121)
µ4
e0
3, e0
3, e0
3, x3
= µ4
x3, e0
4, e0
4, e0
4
= x3, (122)
µ4
e0
4, e0
4, e0
4, x4
= µ4
x4, e0
1, e0
1, e0
1
= x4. (123)
Each of the elements x1, x2, x3, x4 has its triple of 4-ary inverses
x0
1 = (x2, x3, x4), (124)
x0
2 = (x3, x4, x1), (125)
x0
3 = (x4, x1, x2), (126)
x0
4 = (x1, x2, x3). (127)
By analogy with Corollary 4 we now have
Corollary 5. Each polyadic idempotent e0
i of the sandwich higher n-regular k-ary semigroup Ssandw
k
is a local left polyadic identity (neutral element) for xi and a local right polyadic identity for xi−1.
Definition 22. A sandwich higher n-regular polyadic semigroup Ssandw
k is called a sandwich n-
inverse polyadic semigroup, if each element xi ∈ S has a unique n-inverse (k − 1)-tuple x0
i ∈ S(k−1)
(92)–(94).
In search of a polyadic analog of the Lemma 4, we have found that the commutation
of polyadic idempotents does not lead to uniqueness of the n-inverse elements (92)–(94).
The problem appears because of the presence of the middle elements in (91) and, e.g., in
(101) (see the discussion after (103)), while the middle elements do not exist in the previous
formulations of regularity.
4.3. Sandwich Regularity with Generalized Idempotents
Now we using the method of the previous section, we construct a sandwich regularity
containing the generalized polyadic idempotents satisfying (8) instead of (96).
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Definition 23. Sandwich k-ary n-regularity conditions for the sequence of inverses X(k) =
(x1, x2, x3, . . . , xk) with generalized polyadic hmi-idempotents e0
i are defined as
µ
[m]
k
m(k−1)
z }| {
e0
i, . . . , e0
i, xi
= xi, i = 1, . . . , k, (128)
where e0
i are the same as in (91)–(94), but (instead of (96)) now satisfying
e0
i
hmi
= µ
[m]
k
m(k−1)+1
z }| {
e0
i, . . . , e0
i
= e0
i, i = 1, . . . , k. (129)
Definition 24. A k-ary semigroup Sk in which every element belongs to a (not necessary unique)
sequence of inverses X(k) satisfying (91)–(94) and (128)–(129) is called a sandwich n-regular
semigroup with hmi-idempotents.
Example 11. In the lowest case, for the sandwich regular binary semigroup Ssandw
2 with generalized
h2i-idempotents (k = 2, n = 2, m = 2) we have (instead of (22))
x1x2x1x2x1 = x1, (130)
x2x1x2x1x2 = x2. (131)
And h2i-idempotents become
e1 = x1x2, e2 = x2x1, (132)
e3
1 = e1, e3
2 = e2. (133)
In the binary case (k = 2) e1, e2 are called tripotents, and for arbitrary m generalized p-
idempotents, where p = m + 1.
It is obvious that (130) and (131) follow from (22), but not vise versa.
Example 12. For the sandwich 3-regular binary semigroup with 3-idempotents we have (k = 2,
n = 3, m = 2)
x1x2x3x1x2x3x1 = x1, (134)
x2x3x1x2x3x1x2 = x2, (135)
x3x1x2x3x1x2x3 = x3, (136)
and
e1 = x1x2x3, e2 = x2x3x1, e3 = x3x1x2, (137)
e3
1 = e1, e3
2 = e2, e3
3 = e3. (138)
Note that (134)–(136) are different from (101)–(103), because in the latter the multipli-
cation is ternary µ3. In opposition to Example 12 we have
19. Universe 2021, 7, 379 19 of 20
Example 13. For the sandwich regular ternary semigroup Ssandw
3 with h2i-idempotents we obtain
(k = 3, n = 2, m = 2)
µ
[3]
3 [x1, x2, x3, x1, x2, x3, x1, x2, x3, x1, x2, x3, x1] = x1, (139)
µ
[3]
3 [x2, x3, x1, x2, x3, x1, x2, x3, x1, x2, x3, x1, x2] = x2, (140)
µ
[3]
3 [x3, x1, x2, x3, x1, x2, x3, x1, x2, x3, x1, x2, x3] = x3, xi ∈ S, (141)
The three polyadic h2i-idempotents are the same as those in (97)–(99)
e0
1 = µ3[x1, x2, x3], (142)
e0
2 = µ3[x2, x3, x1], (143)
e0
3 = µ3[x3, x1, x2], (144)
but now they satisfy the h2i-idempotence condition (8)
µ
[2]
3
e0
i, e0
i, e0
i, e0
i, e0
i
= e0
i, i = 1, 2, 3. (145)
instead of (100) being the h1i-idempotence (9).
In [36] it was shown that the representatives of a congruence (residue) class [[a]]b form
a k-ary semigroup
R
(a,b)
k =
xa,b(l)
| µk
, xa,b(l) = a + lb ∈ [[a]]b, (146)
l ∈ Z, a ∈ Z+, b ∈ N, 0 ≤ a ≤ b − 1,
with respect to the multiplication
µk[xa,b(l1), xa,b(l2), . . . , xa,b(lk)] = xa,b(l1)xa,b(l2) . . . xa,b(lk) mod b, (147)
if
ak
= a mod b. (148)
The limiting cases a = 0 and a = b − 1 correspond to the binary semigroup R
(0,b)
2
(residue class [[0]]b) and the ternary semigroup R
(b−1,b)
3 (residue class [[b − 1]]b), respec-
tively (for details, see [36]).
Example 14. To construct a concrete realization of the lowest sandwich regularity (139)–(141) we
consider the ternary semigroup of congruence class representatives R
(b−1,b)
3 (the second limiting
case) with the multiplication
µ3[xb−1,b(l1), xb−1,b(l2), xb−1,b(lk)] = xb−1,b(l1)xb−1,b(l2)xb−1,b(l3) mod b, (149)
which is totally commutative. In terms of the representative numbers l the ternary product is
l = l1 + l2 + l3 + 2, l, li ∈ Z, which shows that the idempotents (142)–(144) coincide. The
3-idempotent sandwich regularity relations (130)–(131) become l1 + l2 + l3 + 3 = 0.
Further analysis of the the constructions introduced here and examples of regularity
extensions would be interesting to investigate in more detail, which can lead to new classes
of higher regular and inverse semigroups and polyadic semigroups.
20. Universe 2021, 7, 379 20 of 20
Funding: This research received no external funding.
Acknowledgments: The author is deeply grateful to Vladimir Akulov, Bernd Billhardt, Nick Gilbert,
Mike Hewitt, Mikhail Krivoruchenko, Thomas Nordahl, Vladimir Tkach, Raimund Vogl and Alexan-
der Voronov for numerous fruitful and useful discussions, valuable help and support. After submit-
ting the first version of this manuscript the author received the preprint “Some attempts to formally
generalise inverse semigroups” (unpublished, 2007), where some similar constructions (in a one
relation approach) were considered by Riivo Must to whom the author expresses thankfulness.
Conflicts of Interest: The authors declare no conflict of interest.
References
1. von Neumann, J. On regular rings. Proc. Nat. Acad. Sci. USA 1936, 22, 707–713. [CrossRef]
2. Clifford, A.H.; Preston, G.B. The Algebraic Theory of Semigroups; American Mathematical Society: Providence, RI, USA, 1961;
Volume 1.
3. Ljapin, E.S. Semigroups; American Mathematical Society: Providence, RI, USA, 1968; p. 592.
4. Howie, J.M. An Introduction to Semigroup Theory; Academic Press: London, UK, 1976; p. 270.
5. Grillet, P.A. Semigroups. An Introduction to the Structure Theory; Dekker: New York, NY, USA, 1995; p. 416.
6. Lawson, M.V. Inverse Semigroups: The Theory of Partial Symmetries; World Scientific: Singapore, 1998; p. 412.
7. Petrich, M. Inverse Semigroups; Wiley: New York, NY, USA, 1984; p. 214.
8. Duplij, S. On semi-supermanifolds. Pure Math. Appl. 1998, 9, 283–310.
9. Duplij, S.; Marcinek, W. Higher Regularity And Obstructed Categories. In Exotic Algebraic and Geometric Structures in Theoretical
Physics; Duplij, S., Ed.; Nova Publishers: New York, NY, USA, 2018; pp. 15–24.
10. Duplij, S.; Marcinek, W. Regular Obstructed Categories and Topological Quantum Field Theory. J. Math. Phys. 2002, 43, 3329–3341.
[CrossRef]
11. Duplij, S.; Marcinek, W. Semisupermanifolds and regularization of categories, modules, algebras and Yang-Baxter equation. In
Supersymmetry and Quantum Field Theory; Elsevier Science Publishers: Amsterdam, The Netherlands, 2001; pp. 110–115.
12. Duplij, S.; Marcinek, W. Braid Semistatistics And Doubly Regular R-Matrix. In Exotic Algebraic and Geometric Structures in
Theoretical Physics; Duplij, S., Ed.; Nova Publishers: New York, NY, USA, 2018; pp. 77–86.
13. Duplij, S. Higher braid groups and regular semigroups from polyadic-binary correspondence. Mathematics 2021, 9, 972. [CrossRef]
14. Gilbert, N.D. Flows on regular semigroups. Appl. Categ. Str. 2003, 11, 147–155. [CrossRef]
15. Wazzan, S.A. Flows on classes of regular semigroups and Cauchy categories. J. Math. 2019, 2019, 8027391. [CrossRef]
16. Sioson, F.M. On regular algebraic systems. Proc. Jpn. Acad. 1963, 39, 283–286.
17. Gluskin, L.M. On positional operatives. Soviet Math. Dokl. 1964, 5, 1001–1004.
18. Monk, J.D.; Sioson, F. m-semigroups, semigroups and function representations. Fundam. Math. 1966, 59, 233–241. [CrossRef]
19. Zupnik, D. Polyadic semigroups. Publ. Math. (Debrecen) 1967, 14, 273–279.
20. Post, E.L. Polyadic groups. Trans. Amer. Math. Soc. 1940, 48, 208–350. [CrossRef]
21. Slipenko, A.K. Regular operatives and ideal equivalences. Doklady AN Ukr. SSR. Seriya A. Fiz-Mat i Tekhn Nauki 1977, 3, 218–221.
22. Kolesnikov, O.V. Inverse n-semigroups. Comment. Math. Prace Mat. 1979, 21, 101–108.
23. Duplij, S. Polyadic Algebraic Structures And Their Representations. In Exotic Algebraic and Geometric Structures in Theoretical
Physics; Duplij, S., Ed.; Nova Publishers: New York, NY, USA, 2018; pp. 251–308.
24. Thierrin, G. Sur les éléments inversifs et les éléments unitaires d’un demi-groupe inversif. C. R. Acad. Sci. Paris 1952, 234, 33–34.
25. Vagner, V.V. Generalized groups. Doklady Akad. Nauk SSSR (N.S.) 1952, 84, 1119–1122.
26. Dörnte, W. Unterschungen über einen verallgemeinerten Gruppenbegriff. Math. Z. 1929, 29, 1–19. [CrossRef]
27. Pop, A.; Pop, M.S. On generalized algebraic structures. Creative Math. Inf. 2010, 19, 184–190.
28. Green, J.A. On the structure of semigroups. Ann. Math. 1951, 54, 163–172. [CrossRef]
29. Liber, A.E. On symmetric generalized groups. Mat. Sb. (N.S.) 1953, 33, 531–544.
30. Liber, A.E. On the theory of generalized groups. Doklady Akad. Nauk SSSR (N.S.) 1954, 97, 25–28.
31. Munn, W.D.; Penrose, R. A note on inverse semigroups. Proc. Camb. Phil. Soc. 1955, 51, 396–399. [CrossRef]
32. Schein, B.M. On the theory of generalized groups and generalized heaps. Theory Semigroups Its Appl. 1965, 1, 286–324.
33. Higgins, P.M. Techniques of Semigroup Theory; Oxford University Press: Oxford, UK, 1992; p. 254.
34. Duplij, S. Supermatrix representations of semigroup bands. Pure Math. Appl. 1996, 7, 235–261.
35. Yaglom, I.M. Complex Numbers in Geometry; Academic Press: New York, NY, USA; London, UK, 1968; p. xii+243.
36. Duplij, S. Polyadic integer numbers and finite (m, n)-fields. p-Adic Numbers Ultrametric Anal. Appl. 2017, 9, 257–281. [CrossRef]